Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve.

Abstractly, a hierarchy can be modelled mathematically as a rooted tree: the root of the tree forms the top level, and the children of a given vertex are at the same level, below their common parent.

Abstractly, an array reference is simply a procedure of two arguments: an array and a subscript vector, which could be expressed as ; but many languages provide special syntax like ; similarly an array element update is abstractly something like, but many languages provide syntax like.

Abstractly programmable toolpath guidance began with mechanical solutions, such as in musical box cams and Jacquard looms.

Abstractly, we can identify the chain rule as a 1-cocycle.

Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

Abstractly, 1 / 8-schisma tuning may be considered the analog, among schismatic tunings, of 1 / 4-comma meantone among meantone tunings, as it also has pure interval ratios of 2: 1 and 5: 4, though with much more accurate interval ratios of 3: 2 and 6: 5 ( less than a quarter of a cent off from just intonation ) than its meantone counterpart.

Abstractly, a parametric equation defines a relation as a set of equations.

Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample.

Abstractly, these are the left adjoint and right adjoint, respectively, to the inclusion functors of in D. In addition, the truncation functors fit into a triangle, and this is in fact the unique triangle satisfying the third axiom above:

Abstractly it is assumed that the economy consists only of capitalist production and that the capitalist economy is equal to capitalist society.

For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x < sub > 0 </ sub >,

The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent.

If G is any subgroup of GL < sub > n </ sub >( R ), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group.

They may then be applied in pre-and post-multiplication to the quaternion representation of the coherency matrix, with the usual exploitation of the quaternion exponential for performing rotations and boosts taking a form equivalent to the matrix exponential equations above.

** Matrix exponential, the matrix analogue to the above

Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix ( in fact, special orthogonal ).

The exponential of this is the orthogonal matrix for rotation around axis v by angle θ ; setting c = cos θ / 2, s = sin θ / 2,

A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of n < sup > 2 </ sup >- 1 dimensional Euclidean space.

This approach can be generalized to matrix exponential and other matrix functions since they can be defined as power series.

* If is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion:

For a matrix Lie group the Lie algebra is the tangent space of the identity I, and the commutator is simply = XY − YX ; the exponential map is the standard exponential map of matrices,

A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.

This can be seen from the inverse properties above: operating twice gives the original data in reverse order, so operating four times gives back the original data and is thus the identity matrix.

Finely distributed intermetallics in a ductile matrix yield a hard alloy while coarse structure gives a softer alloy.

Here D is a traceless matrix such that αDe gives the derivative of e with respect to z.

Cells in the epidermis contain a structural matrix of keratin, which makes this outermost layer of the skin almost waterproof, and along with collagen and elastin, gives skin its strength.

Define a data matrix, X < sup > T </ sup >, with zero empirical mean ( the empirical ( sample ) mean of the distribution has been subtracted from the data set ), where each of the n rows represents a different repetition of the experiment, and each of the m columns gives a particular kind of datum ( say, the results from a particular probe ).

The left multiplication of a matrix A with a scalar λ gives another matrix λA of the same size as A.

This highly crystalline metalloid, gives type metal its hardness and a much better and sharper cast from the matrix.

In fact, this expression, ½ (( trA )< sup > 2 </ sup >− tr ( A < sup > 2 </ sup >)), always gives the coefficient c < sub > n − 2 </ sub > of λ < sup > n − 2 </ sup > in the characteristic polynomial of any n × n matrix ; so, for a 3 × 3 matrix A, the statement of the Cayley – Hamilton theorem can also be written as

These relations are a direct consequence of the basic properties of determinants: evaluation of the ( i, j ) entry of the matrix product on the left gives the expansion by column j of the determinant of the matrix obtained from M by replacing column i by a copy of column j, which is if and zero otherwise ; the matrix product on the right is similar, but for expansions by rows.

Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group homomorphism

The decoherence irreversibly converts the " averaged " or " environmentally traced over " density matrix from a pure state to a reduced mixture ; it is this that gives the appearance of wavefunction collapse.

is called the reduced density matrix and gives information about the system only.

In this case, the theorem gives a formula for the Jacobian matrix of the inverse.

Diagonalizing this matrix gives a conjugation of this group with the standard group O ( p, q ).

The following table gives the number of operations in the k-th step of the QR-decomposition by the Householder transformation, assuming a square matrix with size n.

A modulation matrix gives you complete control over your signal flow, letting you modulate anything within Thor.

A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.

Just check that times the RHS of the Woodbury identity gives the identity matrix:

* Lagrange algorithm gives a way to compute an orthogonal basis, and thus compute a diagonal matrix congruent ( thus, with the same signature ) to the other one: the signature of a diagonal matrix is the number of positive, negative and zero elements on its diagonal.